In homological algebra in mathematics, the homotopy category K(A) of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes Kom(A) of A and the derived category D(A) of A when A is abelian; unlike the former it is a triangulated category, and unlike the latter its formation does not require that A is abelian. Philosophically, while D(A) turns into isomorphisms any maps of complexes that are quasi-isomorphisms in Kom(A), K(A) does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence. Thus, K(A) is more understandable than D(A).
Definitions Let A be an additive category. The homotopy category K(A) is based on the following definition: if we have complexes A, B and maps f, g from A to B, a chain homotopy from f to g is a collection of maps h n : A n → B n − 1 {\displaystyle h^{n}\colon A^{n}\to B^{n-1}} (not a map of complexes) such that
f n − g n = d B n − 1 h n + h n + 1 d A n , {\displaystyle f^{n}-g^{n}=d_{B}^{n-1}h^{n}+h^{n+1}d_{A}^{n},} or simply f − g = d B h + h d A . {\displaystyle f-g=d_{B}h+hd_{A}.}
This can be depicted as:
We also say that f and g are chain homotopic, or that f − g {\displaystyle f-g} is null-homotopic or homotopic to 0. It is clear from the definition that the maps of complexes which are null-homotopic form a group under addition. The homotopy category of chain complexes K(A) is then defined as follows: its objects are the same as the objects of Kom(A), namely chain complexes. Its morphisms are "maps of complexes modulo homotopy": that is, we define an equivalence relation
f ∼ g {\displaystyle f\sim g\ } if f is homotopic to g and define
Hom K ( A ) ( A , B ) = Hom K o m ( A ) ( A , B ) / ∼ {\displaystyle \operatorname {Hom} _{K(A)}(A,B)=\operatorname {Hom} _{Kom(A)}(A,B)/\sim }
to be the quotient by this relation. It is clear that this results in an additive category if one notes that this is the same as taking the quotient by the subgroup of null-homotopic maps. The following variants of the definition are also widely used: if one takes only bounded-below (An=0 for n<<0), bounded-above (An=0 for n>>0), or bounded (An=0 for |n|>>0) complexes instead of unbounded ones, one speaks of the bounded-below homotopy category etc. They are denoted by K+(A), K−(A) and Kb(A), respectively. A morphism f : A → B {\displaystyle f:A\rightarrow B} which is an isomorphism in K(A) is called a (chain) homotopy equivalence. In detail, this means there is another map g : B → A {\displaystyle g:B\rightarrow A} , such that the two compositions are homotopic to the identities: f ∘ g ∼ I d B {\displaystyle f\circ g\sim Id_{B}} and
g ∘ f ∼ I d A {\displaystyle g\circ f\sim Id_{A}} . The name "homotopy" comes from the fact that homotopic maps of topological spaces induce homotopic (in the above sense) maps of singular chains.
Remarks Two chain homotopic maps f and g induce the same maps on homology because (f − g) sends cycles to boundaries, which are zero in homology. In particular a homotopy equivalence is a quasi-isomorphism. (The converse is false in general.) This shows that there is a canonical functor K ( A ) → D ( A ) {\displaystyle K(A)\rightarrow D(A)} to the derived category (if A is abelian).
The triangulated structure The shift A[1] of a complex A is the following complex
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